tensorParametrization -- Constructs the morphism that maps a word to its coefficient in the given tensor.

Usage:

m = tensorParametrization(f)
Inputs:
- T, an instance of the type NCRingElement, the tensor encoding the parametrization;
Optional inputs:
- CoefficientRing => a rational number, default value null, the coefficient ring of the domain polynomial ring;
- VarWordTable => ..., default value null, a hash table that assigns variables of a free polynomial ring to words. If provided, the ring is used as the domain of the map;
Outputs:
- m, a ring map, the parametrization encoded by T;

Description

Given some $k$-algebra $A$, a tensor $f = \sum_{\mathtt{w} \in I} a_w \mathtt{w} \in A \langle\texttt{1},\ldots,\texttt{d}\rangle$ can be viewed as a morphism $\mathrm{Spec} \ A \to k^I$. This method constructs the associated map of coordinate rings, that is, the morphism $k[b_w \ | \ w\in I] \to A$ that maps $b_\mathtt{w}$ to $a_\mathtt{w}$. Here $k$ is the $\texttt{baseRing}$ of $A$. If $\texttt{CoefficientRing}$ is provided, it replaces $k$. If $\texttt{VarWordTable}$ is provided, the map above is precomposed with the morphism induced by the assignments in the hash table.

i1 : R = QQ[a,b];

i2 : A2 = wordAlgebra(2, CoefficientRing => R);

i3 : T = a^2 * [1,1]_A2 + a * b * [1,2]_A2 + b^2 * [2,2]_A2

      2   2            2   2
o3 = b Lt  +a*bLt Lt +a Lt
         2       1  2     1

o3 : A2

i4 : tensorParametrization(T); wordFormat T

o4 : RingMap R <-- QQ[b      , b      , b      ]
                       [1, 2]   [1, 1]   [2, 2]

      2                        2
o5 = b  [2, 2] + a*b [1, 2] + a  [1, 1]

i6 : S = QQ[x,y,z];

i7 : vwtable = hashTable({(x,[1,1]),(y,[1,2]),(z,[2,2])})

o7 = HashTable{x => [1, 1]}
               y => [1, 2]
               z => [2, 2]

o7 : HashTable

i8 : m = tensorParametrization(T, VarWordTable => vwtable)

                  2        2
o8 = map (R, S, {a , a*b, b })

o8 : RingMap R <-- S

i9 : kernel m

            2
o9 = ideal(y  - x*z)

o9 : Ideal of S

For a key use example see Computing Path Varieties.

Ways to use tensorParametrization:

tensorParametrization(NCRingElement)

For the programmer

The object tensorParametrization is a method function with options.

The source of this document is in PathSignatures/documentation.m2:918:0.